The Existence of an Optimal Path in a Growth Model with Endogenous Technical Change
We find a growth path B which is not feasible, and which grows exponentially. This path B bounds all feasible paths. Using this bound we define a finite measure on the set of integers, A, and consider a "weighted" Banach space H\ of all sequences which are summable with respect to this measure A. This space includes all bounded sequences, and many exponentially growing sequences; in particular, it contains all the feasible growth paths in our economy. Therefore, without loss of generality, we consider the problem of maximizing a welfare function in the space H\, restricted to the set of all feasible growth paths. For this it would suffice to prove that the set of all feasible paths is compact, and the welfare function is continuous on it in the norm of the space H\. However, these conditions are not satisfied: the logarithmic utility function is undefined over certain paths, and the closed bounded subset of H\ which includes all the feasible growth paths, is not closed and therefore not compact. However, we find a closed subset of the space H\ which contains all feasible paths yielding utility values which exceed a minimal level. We prove that this set is compact and that, when restricted to this set, the welfare function is norm continuous. This set contains an optimum of our problem, thus establishing existence of an optimal growth path with endogenous technical change. The techniques utilized here rely on the use of weighted Lp spaces introduced in Chichilnisky ,  and Chichilnisky and Kalman .
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